State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

TL;DR

This paper develops a framework to construct and analyze fermionic phases of matter and spin-TFTs in 2+1 dimensions using bosonic shadow theories, fermionic anyon condensation, and string-net models, revealing new group structures.

Contribution

It introduces a novel formalism connecting bosonic shadow theories with fermionic phases via fermionic anyon condensation, including new string-net constructions and symmetry classifications.

Findings

Constructed Turaev-Viro-like and Levin-Wen-like models for fermionic phases

Described the group structure of fermionic SPT phases, including the quaternion group

Provided a new perspective on fermionic topological phases using bosonic shadow theories

Abstract

It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic "shadow" theories, which are obtained from the original theory by "gauging fermionic parity". The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by the product of fermion parity and internal symmetry G. The quaternion group makes a surprise appearance.